Adds a list of clickable category links below the main event search bar (sometimes referred to as the "Tribe Bar"). Tested with TEC 4.6.26.

category-links-below-event-search-bar.php

<?php
/**
 * Add a list of clickable category links below the event
 * search bar.
 *
 * Can be easily styled using the following selector:
 *
 *     .the-events-calendar-category-list
 */
add_action( 'tribe_events_bar_after_template', function() {
	$terms = get_terms( [
		'taxonomy' => Tribe__Events__Main::TAXONOMY
	] );

	if ( empty( $terms ) || is_wp_error( $terms ) ) {
		return;
	}

	print "<div class='the-events-calendar-category-list'> <ol>";

	foreach ( $terms as $single_term ) {
		$url = esc_url( get_term_link( $single_term ) );
		$name = esc_html( get_term_field( 'name', $single_term ) );

		print "<li> <a href='$url'>$name</a> </li>";
	}

	print "</ol> </div>";
} );